Symmetry-constrained low-energy effective Hamiltonian for topological RuC and OsC monolayers

We derive a low-energy $\mathbf{k}\cdot\mathbf{p}$ effective Hamiltonian for monolayer osmium carbide (OsC) and ruthenium carbide (RuC) in a planar hexagonal configuration. First-principles calculations indicate that both monolayers are dynamically stable and exhibit features of a two-dimensional quantum spin Hall (QSH) phase, characterized by a nontrivial $\mathbb{Z}_2$ topological invariant. Using symmetry analysis at the $\Gamma$ point, we construct a multiband $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian including spin-orbit coupling and reduce it to a four-band low-energy model through L\"owdin partitioning. The effective Hamiltonian has a block-diagonal form, with two blocks related by time-reversal symmetry, analogous to the Bernevig--Hughes--Zhang (BHZ) model. In contrast to the standard BHZ form, the symmetry-allowed off-diagonal coupling contains quadratic momentum-dependent terms, which modify the low-energy dispersion near the $\Gamma$ point. The fitted parameters reproduce the ab initio band structures in the low-energy region, yielding a compact model for analyzing the electronic and topological properties of monolayer OsC and RuC.